Foundation · Explanatory preview

A local expansion

Near a point, value, slope and curvature describe the leading change. Discarding the remaining terms requires a scale argument.

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Why can a smooth function be replaced by a few terms?

The first derivative is local change per unit of the input. The second derivative describes how that first rate changes. A Taylor expansion writes the nearby value using those rates. The dots stand for a remainder; they are not automatically negligible.

f(x+\Delta)=f(x)+\Delta f'(x)+\frac{\Delta^2}{2}f''(x)+\ldots

Nearby value equals the value plus displacement times slope plus half the squared displacement times curvature, followed by a remainder.

For a diffusion limit, the mean square step shrinks in proportion to the time interval, while the retained variance per unit time stays finite. Higher moments and the variation scale of the density must justify neglecting higher terms. A real tracer does not have independent steps at arbitrarily short times.

One worked example

(x+\Delta)^2=x^2+2x\Delta+\Delta^2

For the square function, the local expansion is exact through the quadratic term.

At x = 2 and Δ = 0.1, the three terms are 4, 0.4 and 0.01, adding to 4.41. Keeping only the value and slope gives 4.4, with an error of 0.01.

A stopping point: Name what is held fixed, which terms are retained, and why the remainder is small.

Extension · Planned callers: Special Relativity §4, Mass–Energy

Binomial Series Expansion for the Relativistic Lorentz Factor γ

In relativistic mechanics, the Lorentz factor γ = (1 − v²/c²)^−1/2 governs time dilation, length contraction, and the relativistic kinetic energy E_k = (γ − 1)mc². Setting x = (v/c)², the function is expanded using Newton’s generalized binomial theorem:

(1 − x)−1/2 = 1 + ½ x + ⅜ x² + ⁵/₁₆ x³ + ³⁵/₁₂₈ x⁴ + …   (|x| < 1)

1. Moderate speeds: partial sums at v/c = 0.6 (x = 0.36)

At sixty percent the speed of light, x = 0.36. The exact value of γ − 1 is 1/√(1 − 0.36) − 1 = 1/0.8 − 1 = 0.25. The progressive partial sums illustrate how higher-order terms accumulate:

Partial sums of γ − 1 at x = 0.36 against exact 0.25
Order	Term added	Term value	Partial sum	Difference to exact (0.25)
1st (linear in x)	½ x	0.180000	0.180000	0.070000
2nd (quadratic)	⅜ x²	0.048600	0.228600	0.021400
3rd (cubic)	⁵/₁₆ x³	0.014580	0.243180	0.006820
4th (quartic)	³⁵/₁₂₈ x⁴	0.0045927	0.247773	0.002227

Notice that the leading quadratic term 0.18 corresponds to the 0.18L mass–energy fixture beside the exact 0.25L, making both the retained classical kinetic energy (½ mv²) and the neglected relativistic remainder visible.

2. Low speeds: catastrophic cancellation and the cancellation-free rule

At everyday velocities such as v/c = 10⁻⁴ (where x = 10⁻⁸), evaluating γ − 1 by naive floating-point subtraction suffers catastrophic loss of precision. Because γ ≈ 1.000000005, subtracting 1 throws away the leading digits and amplifies round-off error.

Comparison at v/c = 10⁻⁴ (x = 10⁻⁸)

First series term (½ x):: 5.0000000000e-9
Cancellation-free route (trustworthy):: 5.0000000375e-9 (computed: 5.0000000375e-9)
Relative diff to first term:: 7.500000e-9 (exact theoretical 0.75x = 7.500000e-9)
Planted naive route 1/√(1−x) − 1:: 5.0000001917e-9 (computed: 5.0000001917e-9)
Naive relative difference:: 3.833145e-8 (approx 3.833e-8; over five times too large)

Cancellation-free evaluation rule: The cancellation-free route usingexpm1(-0.5 * log1p(-x)) or x / (√(1−x) · (1 + √(1−x)))is required for trustworthy numerical evaluation. Naive subtraction in IEEE 754 double precision yields 5.0000001917e-9 instead of the true 5.0000000375e-9, corrupting relativistic corrections with false precision artifacts.

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